Analyzing the Approximation Error of the Fast Graph Fourier Transform

TL;DR

This paper investigates the approximation error distribution of a fast, sparse version of the graph Fourier transform, which balances computational efficiency and accuracy through a truncation parameter.

Contribution

It extends previous work by analyzing how the approximation error varies across different graph types and spectral components, enhancing understanding of the FGFT's accuracy.

Findings

Error distribution varies across spectrum and graph types

Truncation parameter J controls the trade-off between accuracy and efficiency

FGFT significantly reduces computation time and storage compared to exact GFT

Abstract

The graph Fourier transform (GFT) is in general dense and requires O(n^2) time to compute and O(n^2) memory space to store. In this paper, we pursue our previous work on the approximate fast graph Fourier transform (FGFT). The FGFT is computed via a truncated Jacobi algorithm, and is defined as the product of J Givens rotations (very sparse orthogonal matrices). The truncation parameter, J, represents a trade-off between precision of the transform and time of computation (and storage space). We explore further this trade-off and study, on different types of graphs, how is the approximation error distributed along the spectrum.